OPTIMAL ORDERING REGIONS UNDER
JOINT REPLENISHMENT
by
Donald B. Rosenfield
Visiting Associate Professor
M.I.T. Sloan School of Management
Senior Consultant
Arthur D. Little, Inc.
Working Paper #1661-85
May 1985
IP_I
1_
1--s---1)-11__
1.
Introduction
The inventory theory literature contains extensive treatment of
as treatment of joint replenishment
single item policies, as well
(i.e.
treated very thoroughly is the problem of joint
is,
a single set-up cost for two or more items,
demand.
optimal
(r,
What is not
ordering frequency) under deterministric demand.
2]
Ignall
replenishment, that
under stochastic
policies may be complex.
Johnson
3] and Kalin
s) policies under certain conditions where
ordering region.
and that
notes that this problem is difficult,
Silver
6]
[4]
is an optimal
hypothesizes can order regions
conditions of major and minor set-up costs.
identify
under the
There are no definitive
results, however, about the nature of the optimal ordering region
This paper deals with the most basic type of
replenishment:
and
when to order
commonly faced with this problem.
however, will also
for the two items
and
Indeed, pratitioners are
Typically items are ordered when
reached for either item.
set order points when
reaches a given order
some theoretical basis, as
Some pratitioners,
the weighted
level.
sum of stock
This approach has
the development in the remainder
The development demonstrates the nature of
paper
shows.
order
surfaces for a single period.
not shown
Both the theoretical
implications are self-evident.
an order point is
issue in joint
two items under a common set-up charge
independent stochastic demands.
practical
of the
the optimal
While the analogous results are
for multiple periods or the infinite horizon, the single-
period results
should indicate
.
the nature of the solution.
1
11
Based on results from single-item theory, under
and holding costs, when items are ordered,
order up to.
Hence with two independent
ordered, there
issue
is the determination of
it exists)
2.
is a unique vector level
the
linear
penalty
there are unique levels
items, when
(s 1 ,
s2)
items are
to order up to.
surface on the cartesian space
from which it is optimal
to
The
(if
to order.
Analysis
Consider a family of two items with a joint order cost.
Thus
the cost of ordering is
c(x,y) =
k + c x + c2 y
if either x > 0 or y > 0
=
0
otherwise
where
x
=
order for item 1
y
=
order for item 2
k
=
setup cost
The issue
becomes the costs of alternative policies for a given
inventory position (xo,yo).
the
two items
expected costs
order-up-to
of h
For linear holding and penalty costs for
and h 2 and Pi
given an initial
level
and P2 respectively,
inventory level
(x,y) is
+ c2(Y - y0 )
k + C1(x - x 0)
x
+ £ Pl( -X)
l(
)d
+
£ hl(x-
0D
+
)dE
X
p 2 (E-y), 2(E)d
y
)l(
-<
X
+
f
h
2
(Y-E)0
2
-a
2
(E)dE
(xo,
the total
yo) and an
1 and
where
2 are the two demand densities.
If x=xo and Y=YO, then
the same expression with k=O holds.
Let
i.(y)
f
=
where Fi
f
=
(-y)*i(E)dE
y
(1-Fi(E))dt
y
is the cumulative distribution for *i.
For a normal
distribution, for example,
y -u
Gi(Y)
iG( --
=
-
where
pi
=
mean of
Qa
=
standard deviation of
=
Go(Y)
I
i
i
(-y)(E)dE
y
where
O is
a standard normal density function.
Additionally,
Y
-
I
(Y-E)0i(E)dE
=
y -
f
ui+
(i-Y)i(E)dE
Given these expressions, we see that if an order is made to bring
inventories up to the levels x and y, the cost is
k-ClX
+
0
(c
+ (hl
1
-
c2Y
-
hu
+
(C
+ P)G 1 (x)
+
+ hl)X
-
1
2
h2U 2
+ h 2 )Y
(h
2
+
2
)G 2 (y)
3
11~--~-.
~_.~
~11_1_1__~_11~.1^
XI__11_^~
__ I·
.
~ _- - ----
_ .-
(1)
Denoting the function in braces as F(x,y),
convex with a unique minimum F(x*, y*)
and noting that F(x,y)
is
we see that the optimal policy
is
1)
order up to (x*, y*) if x 0
and F(xO,
2)
yO)
or der up to
(x O,
and F(x O,
yO)
or der up to
3)
if x0
F(x O ,
(x*, y)
y*)
y*
x*,
YO
< y*
+ k
< x*,
YO >
Y*
YO ) + k
Do not order otherwise
Conditions 2 and 3 yield the usual
policy
yO <
+ k
>
if x0
yO) > F(x*,
and F(x o ,
4)
F(x*, y*)
y*)
x*,
<
in y*
separable.
and x
respectively,
The interesting case
type of
order points in an (s,
S)
as the function in braces is
s condition 1, which defines the
optimal regions to order when both items are below their optimal
levels.
The optimal
F(xo,
=
yO)
surface
F(x*,
y*)
in the region x 0
two variables x and y,
F along y(x) to be zero.
So
dy
2y
dx
+ aF
=
2x
0
or
F
-d
x
=
d_!
dx
F
From (1),
c.
ci
y
the partials are of the form
+ hi+
i.
(Pi
1
+
is
(2)
isocontours
(2) can be found by the equating the directional
derivative of
aF
yO < y*
+ k
With a continuous function of
of the form
< x*,
h )Gi
i i
4
Furthermore, since
Gi(z)
f
=
(E-z)Ol(e)dE
I (1-Fi(M))dE
=
Z
Gi(z)
=
Z
-
Fi(z)
1
Hence the differential equation for the optimal ordering contour is
d2
=
-
dx
(PL + h 1 )Fl(x)
l
(P2
+h2.)F
2
(Y)
(p
-
(P 2
-
c1)
c2 )
which yields
G 2 (x)(p2
+ h 2)
+ G(x)(p 1
+ h1 )
+ (c 2 + h 2 )Y
+ (c1
+
hl)x = C
(3)
where C is a constant.
The interpretation of (3) is as follows:
The optimal ordering
surface is obtained when the sum of functions of x and y are
constant.
Each function represents the relative "vulnerability" of
the inventory for one of the two products.
minimized upon ordering).
(The functions are
When the total vulnerability reaches a
threshold, an order is placed.
3.
Discussion
This type of relationship typically yields a curve of the form
shown in Figure 1.
When x is near its optimal order point and y is
near its optimal order up to level
function R 2 (y)
=
G 2 (y)(p
2
+ h 2)
+
(the point a in Figure 1) the
(c 2 + h 2 )y is close to its minimum
(that defines the order-up-to level) and is changing very slowly.
the other hand,
Rl(x)
=
Gl(x)(pl + h1 )
+ (c1
On
+ hl)x is changing
rapidly, as x is near its order point and not hear its minimum.
Thus,
in order for (3) to hold, small changes in x must be
accompanied by large changes in y.
The opposite holds at point b.
At point c, both functions R 1 (y) and R 2 (x) are changing at about the
5
same rate
(when scaled) and hence the optimum curve is pitched at 45
in order to maintain
In practice,
difficult,
of
(3).
since the use of
(3) might be computationally
it might make sense to approximate
the curve by the union
two rays and one line segment arising from the three curves
x
=
constant
= x* -
Q
(4a)
y
=
constant
= y*
Q2
(4b)
-
and
Q1 x
where Q
+ Q2
= constant
(4c)
and Q2 are the lot
sizes
(order-up-to-level minus
item order point) for x and y respectively.
order points if
are the
The
joint replenishment did not exist.
ordering region is the union
the curves
x* and y* are the
The constants in 4a and 4b hence
optimal order-up to levels.
individual
single
of three regions
below or to the left of
4a through 4c.
In other words, there
should be separate trigger points for x, y
and a weighted sum of x and y.
This allows more accuracy than a
system with order points for x and y only.
While the curve of
(3)
has not been demonstrated to hold for the infinite horizon or average
cost cases,
there is a theoretical
aggregate sums of
basis for trigger
levels for
stock.
As a pratical matter,
the weighting on the weighted sum curve
should be based on the maximum replensihment quantities for each item
(i.e.
the
lot
hundred unit
sizes Q1
and Q2)-
If,
replenishment amount for
for example,
there is a one-
y and a ten-unit amount for x,
then y should be more heavily weighted in determining the weighted
sum reorder
level.
This
proportional proximity to
type of approach examines the sum of
the individual
6
item order points.
The
constant for the weighted sum constraint
relationship
(3) and is hence the left
either
* -
(x*,
Q2)
or
(x*
is the same as C in
side of
evaluated at
- Q, y*).
Sometimes, when both items are ordered,
smaller
(3)
set-up for the second item.
of what is known as can-order points.
This
there is an additional
leads to the desirability
The same type of
arguments can
be used to show the optimality of the type of policy shown in Figure
2.
The curve of the
oint
order surface between
the unconditional
order and can-order points is based. on the same type of constant
directional
derivative argument.
7
11
References
1.
R. Brown, Decision Rules for Inventory Management,
Rinehart and Winston, NY, 1967.
2.
E. Ignall, "Optimal Continuous Review Policies for Two-product
Inventory Systems with Joint Setup Costs", Management Science,
15,
1969, 278-283.
3.
E. Johnson, "Optimality and Computation of (,
S) Policies in
the Multi-item Infinite Horizon Inventory Problem", Management
Science, 13, No. 7, 1967, 475-491.
4.
D. Kalin, "On the Optimality of (,
s)
olicies", Mathematics
of_Operations Research, 5, No. 2, May 1980, 293-307.
5.
H. Scarf, "The Optimality of (S, s) Policies in the Dynamic
Inventory Problem", Mathematical Methods in the Social
Sciences, K. Arrow, S. Karlin, and P. Suppes (eds,), Stanford
University Press, Stanford, CA, 1960.
6.
E. Silver, "A Control System for Coordinated Inventory
Replenishment", International Journal of Production Research,
22, No. 6, 1974, 647-671.
7.
E. Silver, "Determining Order Quantities in Joint
Replenishment under Deterministic Demand, Management Science,
22, No. 12, 1976, 1351-1361.
8.
E. Silver, "Some Characteristics of a Special Joint-order
Inventory Model", Operations Research, 13, No. 2, March 1965,
319-327.
9.
A. Veinott, Optimal Policy for a Multi-Product Dyanmic
Nonstationary Inventory Problem", Management siet
sence, 12,
Nov. 1965, 206-221.
8
Holt,
No.
3,
Optimal y1
order up to
level for y
Optimal y*
order point
for y alone
Optimal
order point
for x alone
x
Optimal
order up
to level
for x
x2
Figure 1.
Optimal Ordering Surface
9
III
Optimum
order-up-to level
-
F
I
y
I
y
can-order
point
I
I
joint
order
surface
unconditional
y-order
point
I
I
I
I
I
x
x
Can-order
point
Unconditional
x order point
Arrows Represent Optimal Orders
Figure 2.
Optimal ordering with can-order points.
10